where \(\mathinner {|{\Psi (t)}\rangle }\)  is the exact, time-dependent many-body wave function of
the system and \(\hat{H}\)  the nuclear many-boy Hamiltonian. The next step is to
adopt an ansatz for the form of \(\mathinner {|{\Psi (t)}\rangle }\) . In the time-dependent
Hartree-Fock (TDHF) theory, we would enforce that \(\mathinner {|{\Psi (t)}\rangle }\)  remains a
product state of single-particle states at all times (=a Slater
determinant); in the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory, we
would similarly choose that \(\mathinner {|{\Psi (t)}\rangle }\)  must be a Bogoliubov vacuum of the
form (REF ) at all times (in other words: the operators
\(\beta _{\mu }\)  are now time-dependent). The third step is to insert the chosen
ansatz for the wave function into (REF ) and find a method to
determine either the single-particle wave functions \(\varphi _k(t)\)  (TDHF) or
the Bogoliubov matrices \(U(t)\)  and \(V(t)\)  (TDHFB). This step is not as
straightforward as in the static case as shown in [1]}; see also
[2]}, [3]}, [4]}, [5]} for additional
discussions. Here, we only recall the basic formula that are relevant for
fission theory. In TDHF, the application of the variational principle leads to
the following equation,
\(i\hbar \frac{\partial \rho }{\partial t} = \big [ h[\rho (t)], \rho (t) \big ],\) 
